Modelling Host-Parasite Coevolution in Continuous Space

Coevolution can lead to

spatial mosaics of phenotypic and genomic diversity

  • Classically: interspecific spatial correlation \(\implies\) coevolution

    • Not true since other processes can generate correlation, limited dispersal in particular

Two projects

focused on disentangling spatial patterns due to

coevolution, limited dispersal and drift

  • Phenotypic (math)

  • Genomic (slim)

  • Focusing on phenotypic model today

Goals

  • Approximate spatial scales of phenotypic turnover and coadaptation
  • Introduce measures of local adaptation for continuously distributed populations
  • Use results to determine which species is locally adapted
    • As function of relative dispersal abilities and strengths of selection
  • Test accuracy of math using simulations (not slim)
    • Benefit: allows us to relax key model assumptions
    • Cost: mapping simulation parameters to math parameters is hard
      • e.g. no such thing as interaction radius in math model

The model

  • Tracks (co)evolution of mean traits across 2D habitat
  • Drift captured by Gaussian noise in mean trait evolution
  • Dispersal is Gaussian in both species
  • Single trait mediates interaction/coevolution
    • Trait matching/mismatching

Key assumptions
(to be relaxed)

  • Spatially homogeneous:
    • abundance densities
    • additive genetic and phenotypic variances
  • Weak coevolutionary selection

What it looks like

Methods

  • Apply Fourier theory to model to approximate:
    • Spatial (intraspecific) auto-covariance functions
      • Provides notion of scale for within species spatial patterns
    • Spatial (interspecific) cross-covariance function
      • Provides notion of scale for between species spatial patterns

Spatial scales of phenotypic turnover

  • Spatial (intraspp) auto-cov functions are approximately Matérn

    • Matérn comes with built-in notion of spatial scale:
  • \[\xi_S = \frac{\sigma_S}{\sqrt{G_S\psi_S}}\]

  • \(\sigma_S=\) dispersal distance of species \(S=H,P\)

  • \(G_S=\) additive genetic variance

  • \(\psi_S=\) cumulative strength of selection

    • \(\psi_H=A_H-B_H, \ \ \ \psi_P=A_P+B_P\)

Spatial scale of coadaptation?

  • Spatial (interspp) cross-cov function analytically intractable
  • How to measure spatial scale of coadaptation?
  • Plan:
    1. Numerically compute cross-cov function
    2. Fit a widely-employed spatial cov function to theoretical result
    3. Fit same function to simulated results to assess accuracy

Local adaptation

  • Can quantify local adaptation using collocated covariance

  • \(\mathcal{L}_H = -B_H\mathrm{Cov}(\bar z_H,\bar z_P)\)

  • \(\mathcal{L}_P = B_P\mathrm{Cov}(\bar z_H,\bar z_P)\)

  • \(B_H,B_P=\) strengths of biotic selection

  • \(\bar z_H,\bar z_P=\) local mean traits

Local adaptation as function of relative dispersal distances

Testing theory with simulations

  • Individual-based simulations
  • Requires fitness functions (expd offspg num)
    • Theory uses growth rates
    • How to relate?
  • Math! (theory of diffusion-limits in particular)

Mapping parameters

Additive genetic variance

\[G_P\approx \sqrt{\mu_P/(A_P+B_P)}\]

Population density

\[\rho_P\approx \frac{1}{\ln\kappa_P}\Big(r_P-\frac{1}{2}\sqrt{\mu_P(A_P+B_P)}\Big),\]

Growth rate

\[r_P\approx\ln\alpha_P+\pi_{\max}(\iota_P-1)\big(1-e^{-2\pi R_\iota^2\rho_H}\big),\]

Strength of biotic selection

\[B_P\approx\gamma\pi_{\max}(\iota_P-1)\big(1-e^{-2\pi R_\iota^2\rho_H}\big)\]

How well does it do?

  • To evaluate, toss down 20 disks for each spp

    • Radius equal to competition radius for each spp
    • Measure population density and additive genetic var within each disk
    • Average across disks for an estimate
  • Result:

    ##          Parameter Expectation Observation
    ## 1     host density   12.888329   12.000000
    ## 2 parasite density    8.707538    8.050000
    ## 3           host G    8.689459    3.554851
    ## 4       parasite G    6.932665    3.441180